A Nash-Kuiper theorem for isometric immersions in a high codimension

Abstract

This paper is devoted to investigating the isometric immersion problem of Riemannian manifolds in a high codimension. It has recently been demonstrated that any short immersion from an $n$-dimensional smooth compact manifold into $2n$-dimensional Euclidean space can be uniformly approximated by $C^{1,\theta }$ isometric immersions with any $\theta \in (0,1/(n+2))$ in dimensions $n\ge 3$. In this paper, we improve the Hölder regularity of the constructed isometric immersions in the local setting, achieving $C^{1,\theta }$ for all $\theta \in (0,1/n)$ in odd dimensions and all $\theta \in (0,1/(n+1))$ in even dimensions. Moreover, we also establish explicit $C^1$ estimates for the isometric immersions, which indicate that the larger the initial metric error is, the greater the $C^1$ norms of the resulting isometric maps become, meaning that their slope become steeper.